# Difficult Time and Work Solved QuestionAptitude Discussion

 Q. If $A$ and $B$ work together, they will complete a job in 7.5 days. However, if $A$ works alone and completes half the job and then $B$ takes over and completes the remaining half alone, they will be able to complete the job in 20 days. How long will $B$ alone take to do the job if $A$ is more efficient than $B$?
 ✖ A. 20 days ✖ B. 40 days ✖ C. 36 days ✔ D. 30 days

Solution:
Option(D) is correct

Let '$a$' be the number of days in which $A$ can do the job alone.

Therefore, working alone, $A$ will complete $\left(\dfrac{1}{a}\right)^{th}$of the job in a day.

Similarly, let '$b$' be the number of days in which B can do the job alone.

Hence, $B$ will complete $\left(\dfrac{1}{b}\right)^{th}$of the job in a day.

Working together, $A$ and $B$ will complete $\left(\dfrac{1}{a}+\dfrac{1}{b}\right)^{th}$of the job in a day.

The problem states that working together, $A$ and $B$ will complete the job in 7.5 days.

i.e they will complete $\left(\dfrac{2}{15}\right)^{th}$of the job in a day.

Therefore, $\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{2}{15}$........(1)

From the question, we know that if $A$ completes half the job working alone and $B$ takes over and completes the next half, they will take 20 days.

As $A$ can complete the job working alone in 'a' days, he will complete half the job, working alone, in $\dfrac{a}{2}$ days

Similarly, $B$ will complete the remaining half of the job in $\dfrac{b}{2}$ days.

Therefore, $\dfrac{a}{2}+\dfrac{b}{2}=20$

$a+b=40 \text{ or } a=40-b$.....(2)

From (1) and (2) we get,

$\dfrac{1}{40-b}+\dfrac{1}{b}=\dfrac{2}{15}$

$⇒ 600=2b(40−b)$
$⇒ 600=80b−2b^2$
$⇒ b^2−40b+300=0$
$⇒ (b−30)(b−10)=0$
$⇒ b=30$ or $b=10.$

If $b=30$, then $a=40−30=10$ or
If $b=10$, then $a=40−10=30$.

As $A$ is more efficient than $B$, he will take lesser time to do the job alone. Hence $A$ will take only 10 days and $B$ will take 30 days.

Note: Whenever you encounter work time problems, always find out how much of the work can '$A$' complete in a unit time (an hour, a day, a month etc). Find out how much of the work can be completed by '$B$' in a unit time. Then add the amount of work done by $A$ and $B$ to find the total amount of work that will be completed in a unit time.

If '$A$' takes 10 days to do a job, he will do $\left(\dfrac{1}{10}\right)^{th}$of the job in a day.

Similarly, if $\left(\dfrac{2}{5}\right)^{th}$ of the job is done in a day, the entire job will be done in $\dfrac{5}{2}$ days.

Edit:  For a different approach of solving the question, please check the comment from RUPESH

## (5) Comment(s)

Arjun
()

A man started a job. Starting from the second day, each day a new man joined with which the capacity of each man doubled. The job was completed in 6 days. On which day will the job be completed if the joining of a new man on a day results in each man working at thrice the rate as he did on the previous day?

1. 6th

2. 5th

3. 4th

4. 3rd

RUPESH
()

$\dfrac{1}{A} + \dfrac{1}{B} = \dfrac{1}{7.5}$

I.E., $\dfrac{1}{A} + \dfrac{1}{B}= \dfrac{1}{15/2}= \dfrac{2}{15}$

IF $A$ IS MORE EFFICIENT THAN $B$, WE CAN'T DIVIDE $\dfrac{1}{15}$ AND $\dfrac{1}{15}$

SO MULTIPLY BY 2 IN NUMERATOR AND NUMERATOR WE GET $\dfrac{4}{30}$

NOW WE HAVE $\dfrac{1}{A}= \dfrac{3}{30}$ AND $\dfrac{1}{B}= \dfrac{1}{30}$

$\dfrac{1}{A}=\dfrac{1}{10}$ IT DOES HALF JOB SO $\dfrac{2}{10} =5$ DAYS

$\dfrac{1}{B} = \dfrac{1}{30}$ IT ALSO DOES REM HALF JOB $\dfrac{2}{30} = \dfrac{1}{15}=15$ DAYS

IF $B$ DOES HALF JOB IN 15 DAYS THEN $B$ DOES FULL JOB IN 30 DAYS

I.E.,$\dfrac{1}{2}= 15$, $1= 30$

THUS, ANSWER IS $\textbf{30 DAYS}$

Chirag Goyal
()

You are using hit and trial method by dividing their efficiency in $3:1$

It's good trick otherwise.

Shreya
()

well 3:1 ratio not given so not accurately we can use this method

Sujit Kumar
()

Hashmi bhai kuch important questions ka trick batawoge..